SUO: Re: CG: Re: Ontology
Jean-Luc,
Most working mathematicians assume an implicit
Platonism, in which the structures they talk about,
think about, and publish papers about are assumed
to exist in some Heaven of Platonic Forms.
A typical (extreme?) example is the famous mathematician
Paul Erdos, who used to say "God has the Big Book, the
beautiful proofs of mathematical theorems are listed there."
He said that when he died his most fervent hope was to
have a look in God's big book.
JLD> There must be of course some underlying metaphysical
> position but it is remarkably "lightweight", does not need
> to be mentionned in the course of explanations and seems
> to suit very well the pragmatic purposes we all have
> (mostly...) when doing mathematics.
The Platonic metaphysics of most mathematicians,
including Erdos, Goedel, and many others, is certainly
not a "lightweight" ontology. It is a very big, very heavy
assumption (God's book would be larger than the universe
if it were printed on paper). But it is an assumption
that many mathematicians (including me) find very appealing.
However, you can do a lot of mathematics without talking
about it -- you just assume it, and go on with business.
> Though the Pikowsky way looks very appealing it still
> uses natural language and, what I was wondering about,
> is whether this can be pushed a bit further and whether
> this "approach" can be fully formalised and still keep
> the intelligibility it has.
That would be very difficult to formalize, because many,
if not most, mathematical insights are based on metaphors
and analogies that range over an open-ended number of
topics. The insights come first, and only much later
are they written down in formal axioms, definitions,
and proofs. The popular presentations, such as the
World of Mathematics, are written last -- they summarize
and popularize the fun part -- the metaphors that lead
to the insights. But they are only written after all
the hard work has been done to verify the insights
definitions, axioms, theorems, and proofs.
CLCE, as I desribe it on my web site, is "syntactic sugar"
for first-order logic. It is useful as a notation for
making the boring part of mathematics more readable
while still being precise. But the really exciting
insights and intuitions that lead to the discovery
of new ways of thinking are very hard to describe
even in unrestricted natural language. That's
inevitable, because there are no words for a
concept that nobody has ever thought before.
The formal proofs are the routine housekeeping
that has to be done after the exciting insights
have been discovered. Erdos, by the way, spent
most of his time doing the fun part, and he let
his co-authors write up and publish the proofs.
Erdos had 507 co-authors -- more than any other
mathematician in history. See
http://www.cs.newcastle.edu.au/~rino/erdos.html
John